International Journal of Analysis and Applications Volume 17, Number 6 (2019), 917-927 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-917 SHEHU TRANSFORM AND APPLICATIONS TO CAPUTO-FRACTIONAL DIFFERENTIAL EQUATIONS RACHID BELGACEM1, DUMITRU BALEANU2,3,∗, AHMED BOKHARI1 1Department of Mathematics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University of Chlef, Algeria 2Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, TR-06530 Ankara, Turkey 3Institute of Space Science, R-077125 Măgurle-Bucharest, Romania ∗Corresponding author: dumitru@cankaya.edu.tr Abstract. In this manuscript we establish the expressions of the Shehu transform for fractional Riemann- Liouville and Caputo operators. With the help of this new integral transform we solve higher order fractional differential equations in the Caputo sense. Three illustrative examples are discussed to show our approach. 1. Introduction One of the most effective methods to solve differential equations is to use integrals transformation. The main advantage of this method is that it transforms the differential problem to an algebraic problem. We recall that the Laplace’s transformation which is widely used to solve differential and integral equations. The Sumudu transform was first defined in 1993 by Watugala who used it to solve engineering control problems [17]. Received 2019-05-24; accepted 2019-07-11; published 2019-11-01. 2010 Mathematics Subject Classification. 26A33, 65R10, 44A20 . Key words and phrases. Shehu transform; Caputo derivative; Mittag-Leffler function. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 917 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-917 Int. J. Anal. Appl. 17 (6) (2019) 918 The Shehu transform was introduced recently by Shehu Maitama and Weidong Zhao [16] and it is a generalization of the Laplace and the Sumudu integral transforms. The authors have used it to solve ordinary and partial differential equations [16]. The Shehu transform is obtained over the set A by [16] : A = { f (t) : ∃N,η1,η2 > 0, |f (t)| < N exp ( |t| ηi ) , if t ∈ (−1)i × [0,∞) } . (1.1) by H [f (t)] = V (s,u) = ∫ ∞ 0 exp ( − st u ) f (t) dt. (1.2) Obviously, the Shehu transform is linear as the Laplace and Sumudu transformations. Theorem 1.1. [16] If the function f(n)(t) is the nth derivative of the function f(t) ∈ A, then its Shehu transform is defined by H [ f(n)(t) ] = (u s )−n V (s,u) − n−1∑ k=0 (u s )(k+1)−n f(k) (0) , n ≥ 1. (1.3) Some properties of Shehu transform are given in [16]. For our results we need some other definitions and some properties. Definition 1.1. A generalization of the exponential function is given by [10] Eα (z) = ∞∑ k=0 zk Γ (αk + 1) , α ∈ C, Re (α) > 0. (1.4) A generalization of Mittag-Leffler function Eα (z) is defined as follows [18]: Eα,β (z) = ∞∑ k=0 zk Γ (αk + β) , α,β ∈ C, Re (α) , Re (β) > 0. (1.5) A generalization of Mittag-Leffler function Eα,β (z) of (1.5) is introduced by Prabhakar [14], as follows: E γ α,β (z) = ∞∑ n=0 γk Γ (αk + β) zk k! , α,β,γ ∈ C, Re (α) , Re (β) > 0, Re (γ) > 0. (1.6) where γk denotes the familiar Pochhammer symbol. Lemma 1.1. [6] In the complex plane C, for any Re (α) , Re (β) > 0, Re (γ) > 0 and ω ∈ C. S ( tγ−1E γ α,β (ωt α) ) = uβ−1 (1 −ωuα)−γ . (1.7) Corollary 1.1. Sumudu transform of Mittag-leffer function Eα,β (z) = ∑∞ n=0 zn Γ(nα+β) , α,β ∈ C, Re (α) , Re (β) > 0 exists and given by S ( tγ−1Eα,β ( ωtβ )) = uγ−1 ( 1 −ωuβ )−1 . (1.8) Int. J. Anal. Appl. 17 (6) (2019) 919 Definition 1.2. Let f ∈ L1 (a,b) . If α ≥ 0, then left sided Riemann–Liouville fractional integral of order α is defined by [1, 12, 13] Iα0+f (t) = 1 Γ (α) t∫ 0 (t− τ)α−1 f (τ) dτ = 1 Γ (α) tα−1 ∗f (t) , α > 0, t > 0, (1.9) I0f (t) = f (t) . Definition 1.3. Let f ∈ L1 (a,b) , and m−1 < α ≤ m. The Caputo fractional derivative of order α (α > 0) is defined as [1, 5, 12] CDα0+f (t) =   1 Γ(m−α) t∫ 0 (t− τ)m−α−1 f(m) (τ) dτ, m− 1 < α ≤ m. ∂m ∂tm f (t) if α = m. Remark 1.1. [13] Under the terms of the previous definition, we have CDα0+f (t) = 1 Γ (m−α) tm−α−1 ∗f(m) (t) . (1.10) Lemma 1.2. [9, Lemma 2.22 p.96] If f(t) ∈ ACn [a,b] or f(t) ∈ Cn [a,b] , then ( IαC0+ D α 0+ ) f(t) = f(t) − n−1∑ k=0 f(k) (0) k! tk. (1.11) As the next theorem shows, the Shehu transform is closely connected with the Sumudu transform, Theorem 1.2. [3] Let f(t) ∈ A with Sumudu transform G (u). Then the Shehu transform V (s,u) of f(t) is given by V (s,u) = u s G (u s ) . (1.12) Lemma 1.3. [15, p. 140-141] If f ∈ L1 (a,b) for any b > a and of exponential order. Then Iα0+f (t) also of exponential order. 2. Main result In this section, we present some results on the transformation of Shehu, as a complementary result of what can be seen in [16]. Theorem 2.1. Let a ∈ C∗ and let f (at) ∈ A. If V (s,u) denote the Shehu transform of f. Then H (f (at)) = 1 a V (s,au) . Int. J. Anal. Appl. 17 (6) (2019) 920 Proof. Using the definition of Shehu transform Eq.(1.1), we get H (f (at)) = ∫ ∞ 0 exp ( − s u t ) f (at) dt. If we set τ = at (t = τ/a), then H (f (at)) = 1 a ∫ ∞ 0 exp ( − s au τ ) f (τ) dτ = 1 a V (s,au) . � Theorem 2.2. Let a ∈ C∗ and let f (t) ∈ A with Shehu transform V (s,u) . Then H ( eatf (t) ) = V (s−au,u) . Proof. Using Eq.(1.2), we have H ( eatf (t) ) = ∫ ∞ 0 exp ( at− s u t ) f (t) dt = ∫ ∞ 0 exp ( − s−au u t ) f (t) dt. By setting s′ = s−au, we get H ( eatf (t) ) = ∫ ∞ 0 exp ( − s′ u t ) f (t) dt = V (s′,u) = V (s−au,u). � Theorem 2.3. For x > 0, the Shehu transform of tx−1 is V (s,u) = Γ (x) (u s )x . (2.1) Proof. For x > 0, the Gamma function is defined by Γ (x) = ∫ ∞ 0 τx−1e−τdτ. If we set τ = s u t ( t = u s τ ) , then we have Γ (x) = ∫ ∞ 0 (s u t )x−1 e− s u t s u dt = (s u )x ∫ ∞ 0 tx−1e− s u tdt = (s u )x H ( tx−1 ) . Then, H ( tx−1 ) = Γ (x) ( u s )x . � Int. J. Anal. Appl. 17 (6) (2019) 921 Lemma 2.1. In the complex plane C, for any Re (α) , Re (β) > 0, Re (γ) > 0 and ω ∈ C. Shehu transform of E γ α,β (ωt α) is given by H ( tβ−1E γ α,β (ωt α) ) = (u s )β ( 1 −ω (u s )α)−γ . (2.2) Proof. Using Eqs.(1.7), (1.12), we get H ( tβ−1E γ α,β (ωt α) ) = (u s )(u s )β−1 ( 1 −ω (u s )α)−γ = (u s )β ( 1 −ω (u s )α)−γ . � Corollary 2.1. Shehu transform of Mittag-Leffler function Eα,β (z) = ∑∞ k=0 zn Γ(αk+β) , α,β ∈ C, Re (α) , Re (β) > 0 exists and given by H ( tβ−1Eα,β (ωt α) ) = (u s )β ( 1 −ω (u s )α)−1 . (2.3) Proof. Using Eq.(2.2) and since Eα,β (z) = E 1 α,β (z) , we get the desired result. � The next Theorem shows the Shehu transform convolution theorem. Theorem 2.4. Let f(t) and g(t) be in A, having Shehu transforms V (s,u) and W (s,u), respectively. Then the Shehu transform of the convolution of f and g (f ∗g) (t) = ∫ ∞ 0 f (t) g (t− τ) dτ, (2.4) is given by H ((f ∗g) (t)) = V (s,u) W (s,u) . (2.5) Proof. First, recall that the Sumudu transform of f ∗g is given by [2] S ((f ∗g) (t)) = uF(u)G(u). (2.6) where F(u) and G(u), are the Sumudu transforms of f(t) and g(t) respectively. Now, since, by the relation (1.12), H [v (t) ∗w (t)] = u s S [v (t) ∗w (t)] = (u s )2 F (u s ) G (u s ) = (u s ) F (u s ) × (u s ) G (u s ) = V (s,u) W (s,u) . � Int. J. Anal. Appl. 17 (6) (2019) 922 Theorem 2.5. Let f satisfy the conditions of Lemma 1.3. Then the Shehu transform of Iαt f (t) exists and given by H (Iαt f (t)) = (u s )α V (s,u) . (2.7) Proof. Since by equation Eq.(1.9) above, Iα 0+ f (t) = 1 Γ(α) tα−1 ∗ f (t) , then by Theorems 2.3 and Theorem 2.4, we have, H (Iαt f (t)) = 1 Γ (α) H ( tα−1 ) H (f (t)) = 1 Γ (α) Γ (α) (u s )α V (s,u) = (u s )α V (s,u) . � Theorem 2.6. If f ∈ ACn (a,b) for any b > a and of exponential order. Then H ( CDα0 f (t) ) = (s u )α V (s,u) − n−1∑ k=0 (s u )α−(k+1) f(k) (0) . (2.8) Proof. Since ( IαC 0+ Dα0+ ) f(t) = f(t) − ∑n−1 k=0 f(k)(0) k! tk, by Lemma 1.2, we have H (( IαC0+ D α 0+ ) f(t) ) = H ( f(t) − n−1∑ k=0 f(k) (0) k! tk ) , thus (u s )α H ( CDα0+f(t) ) = V (s,u) − n−1∑ k=0 (u s )k+1 f(k) (0) , finally, we get H ( CDα0+f(t) ) = (u s )−α V (s,u) − n−1∑ k=0 (u s )k+1−α f(k) (0) . By other method, we can use Eq. (1.12), and that [8] S ( CDα0 f (t) ) = u−α ( G (u) − n−1∑ k=0 ukf(k) (0) ) . In fact, H ( CDα0+f(t) ) = (u s )(u s )−α ( G (u s ) − n−1∑ k=0 (u s )k f(k) (0) ) = (u s )1−α (s u V (s,u) − n−1∑ k=0 (u s )k f(k) (0) ) = (u s )−α ( V (s,u) − n−1∑ k=0 (u s )k+1 f(k) (0) ) . Int. J. Anal. Appl. 17 (6) (2019) 923 By anothor method, we have by Remark 1.1, CDα 0+ f (t) = 1 Γ(n−α)t n−α−1 ∗f(n) (t) , n− 1 < α ≤ n, then by using Theorems 2.3 and Theorem 2.4, we obtain H ( CDα0+f(t) ) = H ( 1 Γ (n−α) tn−α−1 ∗f(n) (t) ) = 1 Γ (n−α) H ( tn−α−1 ) H ( f(n) (t) ) = 1 Γ (n−α) Γ (n−α) (u s )n−α [(s u )n V (s,u) − n−1∑ k=0 (s u )n−(k+1) f(k) (0) ] = (u s )−α [ V (s,u) − n−1∑ k=0 (u s )k+1 f(k) (0) ] . � 3. Applications We take into consideration a general linear ordinary differential equation with fractional order as follows: CDα0+y(t) = n∑ i=1 biy (i) (t) + g (t) , n− 1 < α ≤ n (3.1) subject to the initial condition y(i) (0) = ai, i = 0, ...,n− 1, (3.2) where ai,bj ∈ R, g (t) ∈ A. When we get Shehu transform of (3.1) taking into consideration (1.3) and (2.8), we obtain Shehu transform of (3.1) as follows H ( CDα0+y(t) ) = H ( n∑ i=1 biy (i) (t) + g (t) ) , By the linearity of Shehu transform, we have H ( CDα0+y(t) ) = n∑ i=0 biH ( y(i) (t) ) + H (g (t)) , = b0y (t) + n∑ i=1 biH ( y(i) (t) ) + H (g (t)) Using Eqs.(1.3), (2.8), we obtain (u s )−α V (s,u) − n−1∑ k=0 (u s )k+1−α y(k) (0) = b0V (s,u) + n∑ i=1 bi [(u s )−i V (s,u) − i−1∑ k=0 (u s )k+1−i y(k) (0) ] + H (g (t)) Int. J. Anal. Appl. 17 (6) (2019) 924 (u s )−α V (s,u) − n∑ i=0 bi (u s )−i V (s,u) = n−1∑ k=0 ak (u s )k+1−α − n∑ i=1 bi i−1∑ k=0 ak (u s )k+1−i + H (g (t)) V (s,u) = ((u s )−α − n∑ i=0 bi (u s )−i)−1 (n−1∑ k=0 ak (u s )k+1−α (3.3) − n∑ i=1 bi i−1∑ k=0 ak (u s )k+1−i + H (g (t)) ) . Operating the inverse Shehu transform on both sides of Eq. (3.3), we get the solution of Eq. (3.1) as follows: y (t) = H−1  ((u s )−α − n∑ i=0 bi (u s )−i)−1 (n−1∑ k=0 ak (u s )k+1−α (3.4) − n∑ i=1 bi i−1∑ k=0 ak (u s )k+1−i + H (g (t)) )] . Example 3.1. When n = 1,b0 = −1 and b1 = g (t) = 0 , we obtain [11] CDαy (t) + y (t) = 0, 0 < α ≤ 1, t > 0, (3.5) with initial condition y (0) = 1. (3.6) Substituting n,b0,b1 and g in (3.4), we get : y (t) = H−1  ((u s )−α − 1∑ i=0 bi (u s )−i)−1 (u s )1−α . y (t) = H−1 [(u s )( 1 − (−1) (u s )α)−1] Thus, by Eq.(2.3), we have V (s,u) = H (Eα (−tα)) . (3.7) When we get the inverse Sumudu transform of (3.7) , we find exact solution of Eq.(3.5) as follows: y (t) = Eα (−tα) . Example 3.2. Below we give the following particular example, which where debate in the literature and here important application in several world problems. Consider the Bagley-Torvik equation [7] D2y (t) +C D3/2y (t) + y (t) = t + 1, (3.8) Int. J. Anal. Appl. 17 (6) (2019) 925 with the initial conditions y (0) = y′ (0) = 1. (3.9) In this case, we have n = 2,b0 = b2 = −1,b1 = 0 and g (t) = t + 1. Applying Eq.(3.4), we get y (t) = H−1  ((u s )−3/2 − 2∑ i=0 bi (u s )−i)−1 ( 1∑ k=0 ak (u s )k+1−3/2 (3.10) − 2∑ i=1 bi i−1∑ k=0 ak (u s )k+1−i + u s + (u s )2)] . Then, y (t) = H−1 (( u s )−1/2 + ( u s )1/2 + ( u s )−1 + 1 + u s + ( u s )2( u s )−3/2 + ( u s )−2 + 1 ) = H−1 (( u s )−1 + ( u s )−1/2 + u s( u s )−2 + ( u s )−3/2 + 1 + 1 + ( u s )1/2 + ( u s )2( u s )−2 + ( u s )−3/2 + 1 ) = H−1 ( u s + (u s )2) . (3.11) Taking the inverse Shehu transform of Eq. (3.11), yields y (t) = t + 1, which is the exact solution. Example 3.3. Consider the following homogeneous fractional ordinary differential equation : [4] CD 1 2 y (t) + y (t) = t2 + Γ (3) Γ ( 5 2 )t32 , t > 0, (3.12) with initial condition y (0) = 0. (3.13) In order to find exact solution of (3.12), we apply Eq.(3.4) , for n = 1, b0 = −1,b1 = 0 and g (t) = t2 + Γ(3) Γ( 52 ) t 3 2 , we obtain y (t) = H−1 ((u s )−1 2 + 1 )−1 ( 2 (u s )3 + Γ (3) (u s )5 2 ) = H−1  2 (us)3 + Γ (3) (us)52( u s )−1 2 + 1   = H−1 ( 2 (u s )3 ) . When we take the inverse Shehu transform of 2 ( u s )3 , we get the analytical solution of Eq.(3.12) y (t) = t2. Int. J. Anal. Appl. 17 (6) (2019) 926 4. Conclusions In the field of fractional calculus finding a new integral transform for solving the ordinary od partial fractional differential equations it is always useful. In this manuscript, the newly suggested Shehu integral transform was applied to solve hgher order fractional differential equations with Caputo derivative. We show the efficiency and and high accuracy of the suggested integral transform. References [1] D. Baleanu, K. Diethelm, E. Scalas, and J. J. 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